MA30056: Complex Analysis. Revision: Checklist & Previous Exam Questions I

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1 MA30056: Complex Analysis Revision: Checklist & Previous Exam Questions I Given z C and r > 0, define B r (z) and B r (z). Define what it means for a subset A C to be open/closed. If M A C, when is M said to be open/closed in A? Define what it means for a sequence (z n ) n N C to be convergent. State the Bolzano-Weierstrass and Heine-Borel theorems, and state a result concerning the completeness of (C, ). Define a path, and define what it means for a path : [a, b] C to be simple, closed and simple-closed, respectively. When is C said to be a (simple/closed/simple-closed) Jordan curve? When is A C said to be path-connected? Define a domain D C. State the Jordan curve theorem. When is a domain D C said to be simply connected? Show that if : [a, b] C is a path and [α, β] [a, b], then [α,β] : [α, β] C is also a path. Define the composition of two paths and show that it is a path. Define the inverse of a path. When is a path : [a, b] C said to be smooth? When is a path : [a, b] C said to be regular/piecewise smooth/piecewise regular? When is C said to be a (simple/closed/simple-closed) contour? Let : [a, b] C be a regular path and = ([a, b]). If h : [ã, b] [a, b] is surjective and regular, show that := h : [ã, b] C is a new regular parametrisation of. If : [a, b] C is a regular path, define the length of = ([a, b]). Hence define the length of a contour. Define what it means for a function f : D C to be continuous on D C. Define what it means for f : C D C to be uniformly continuous on D. Let f : D C be defined on a domain D C. Define what it means for f to be complex differentiable at z D. Define what it means for h to complex differentiable/holomorphic on D. Let f : C D C be holomorphic and write u = Re f, v = Im f. Show that the first-order partial derivatives u x, u y, v x and v y of u and v exist on D and satisfy u x = v y, u y = v x in D, and that f (z) = u x (z) + iv x (z) = v y (z) iu y (z), z D. Let f : C D C be holomorphic in D with f 0 in D. Show that f is constant on D. 1

2 Let f = u + iv : C D C be holomorphic. Show that u, v : D R are harmonic functions and that ( ) 0 1 u = v in D. 1 0 Let u, v : R 2 D R be continuously differentiable on D and satisfy u x = v y, u y = v x on D. Show that f = u + iv : D C is holomorphic, using the fact that f must be real differentiable in D, viewed as a subset of R 2 or of C as convenience dictates. 1.) 2007 Exam (Question 1) (c) Let ϕ : D R be twice continuously differentiable and harmonic. Show that f = ϕ x i ϕ y is holomorphic. (d) Show that every harmonic function ϕ : D R, where D is simply connected, is the real part of a holomorphic function. For your proof you may assume that: (i) every holomorphic function f : D C has an anti-derivative F : D C; (ii) if u x = ϕ x and u y = ϕ y for u, ϕ : D R, then ϕ = u + c for some c R. (e) Let ϕ(x, y) = x 2 y 2 ; verify that ϕ is harmonic and find a holomorphic function F with ϕ = Re F. 2.) 2005 Exam (Question 1) (d) Let u(x, y) = x y. Find a holomorphic function f : C C with u = Re f. 3.) 2008 Exam (Question 1) (f) Let u(x, y) = sin(x) cos(y). Is there a holomorphic function f : C C with u = Re f? If so, say which function f; if not, give a reason. Let f : D C be continuous and let : [a, b] D be a piecewise regular path. Define the path integral f(z), and state why this is well-defined. Show that the path integral has the property of parameter invariance: that is, show that if : [a, b] D and = h : [ã, b] D are two parametrisations of a contour = ([a, b]), then f(z) = ± f(z). 2

3 What is the relationship between f(z) and f(z)? Read the subsection on the orientation of a contour. Let f : ([a, b]) C be continuous, and denote M = sup f(z), L = z ([a,b]) b a (t) dt. Show that M < and f(z) ML. Let f : D C be continuous on D and suppose that f has an anti-derivative F : D C (so that F is holomorphic in D with F = f). Show that for any piecewise regular path : [a, b] D, it follows that f(z) = F ((b)) F ((a)), and for every closed contour D, it follows that f(z) = 0. Let f : D C be holomorphic in a domain D C and let D be a simple closed contour such that I D. State Cauchy s theorem for f and. State Cauchy s theorem for simply connected domains. Let f : D C be continuous on a domain D and assume that f(z) = 0, for every closed contour D. Show that f has an anti-derivative F : D C. Hence, show that if f is holomorphic on a simply connected domain D, then f has an anti-derivative F : D C. State a weak version of Cauchy s theorem. Prove this version of Cauchy s theorem by using Green s Theorem. Define a triangle and state the Cauchy-Goursat theorem. Define a star-domain. Prove if f : D C is holomorphic on a star-domain D C, then f has an anti-derivative F : D C. State and prove Cauchy s theorem for star-domains. 3

4 State the Homotopy Version of Cauchy s theorem, and give a generalisation of this theorem to multiple simple closed contours. Evaluate z =2 z 2 1, by using the generalisation of the Homotopy Version of Cauchy s theorem. Let f : D C be holomorphic on a domain D and let D be a simple closed contour such that I D. Show that, for all z I, f(z) = 1 f(w) 2πi w z dw. Let f : D C be holomorphic and let D be a simple closed contour such that I D. Show that f (n) (z) = n! f(w) 2πi (w z) dw, z I n+1, n N 0. (State any result you are using.) State Morera s theorem. Let f : D C be holomorphic and suppose that, for z 0 C, the circle B R (z 0 ) is such that B R (z 0 ) D. Show that for any n N, f (n) (z 0 ) n! R n max z f(z). Define an entire function, and show that any bounded entire function is constant (Liouville s Theorem). State Gauss Fundamental Theorem of Algebra. Let f : B R (z 0 ) C be holomorphic, and assume that f(z) f(z 0 ), z B R (z 0 ). Show that f is constant in the disk B R (z 0 ). What is a logarithm on a simply connected domain D? What is the principal value (of the logarithm) Log(z) of the number z? Define the power a b for complex numbers a, b. 4.) 2007 Exam (Question 2) (d) Let f : D C be holomorphic and r > 0 so that the closed disk B r (z 0 ) D; assume that min z z0 =r f(z) > f(z 0 ) > 0. Show that f has a zero in B r (z 0 ). (Hint: suppose not and consider g(z) = 1 ). f(z) 4

5 5.) 2005 Exam (Question 4) (e) Let f be an entire function and suppose that f(z) e Re z for all z C. Prove that there exists a c C with c 1, such that f(z) = c e z for all z C. 6.) 2008 Exam (Question 3) (e) Show that f(z) = e i z is not an entire function. Justify all your claims. (f) Suppose that f is entire and that f(z) z 2 for all sufficiently large values of z, say, for z r. Prove that f must be a polynomial of degree at most 2. (Hint: Use Cauchy s Inequalities for the Derivatives to conclude that f (3) vanishes.) 7.) 2005 Exam (Question 2) (c) Let f(z) = 1. 1+z 2 (i) Show that there is a closed regular path : [a, b] C \ {±i} with f(z) 0. (ii) Deduce that f cannot have an anti-derivative in C \ {±i}. (d) Let f(z) = 4z 4 and a > b > 0. Compute the path integral f(z), where : [0, 1] C, t (t) = a sin πt ( 2 + ib 1 cos πt ). 2 Additional question: What is the catch in (d)? 8.) 2005 Exam (Question 3) (d) Let := {z = x + iy C max{ x, y } = 3} (oriented in the anti-clockwise sense). Evaluate e z (i) (z (2 iπ/2)), (ii) e z 2 + 4i + z, (iii) cos(πz). 1 z 9.) 2008 Exam (Question 2) (c) Show: (i) f(z) = 1 has an anti-derivative in the cut plane C \ R z 0, but (ii) f(z) = 1 has no anti-derivative in the punctured plane C \ {0}. z State any result you are using to establish these claims. (e) Is there a logarithm of 0? Justify your answer. (f) Calculate all possible values of i i. What is the principal value of i i? 5

6 Define what it means for the sequence of functions {f n } n N (where f n : D C) to converge uniformly to the function f : D C on D. State the Cauchy criterion for uniform convergence, in this context. Define what it means for a sequence of functions {f n } n N to converge locally uniformly to a function f on D. Show that the locally uniform limit of continuous functions is continuous. Show that if : [a, b] C is piecewise regular, = ([a, b]), and f : C is the uniform limit of a sequence of continuous functions f n : C, then f(z) = lim f n (z). n Let {f n } n N (where f n : D C for all n) be a sequence of holomorphic functions converging locally uniformly to a function f : D C on D. Show that f is holomorphic, and that f n f locally uniformly on D, as n. State the Weierstrass M-test for a sequence of functions {f k } k N0. Show that functions defined by power series are holomorphic within the disk of convergence, and that the derivative is obtained by term-by-term differentiation. Let f : D C be holomorphic and suppose that B R (z 0 ) D for some z 0 D and R > 0. Show that for any z B R (z 0 ), f(z) = k=0 a k (z z 0 ) k, where a k = f (k) (z 0 ), k N 0. k! Define what it means for a function f : D C to be analytic in D. Let f(z) = a k (z z 0 ) k be a power series with radius of convergence R > 0 k=0 (or R = ). Suppose that f(z n ) = 0, n N, for some sequence (z n ) n N B R (z 0 ) \ {z 0 } such that z n z 0 as n. Show, by induction, that a k = 0, k N 0 (that is, f 0). Hence show that the power series expansion of a function is unique. Suppose that f : D C is holomorphic and satisfies f(z n ) = 0, n N, for some sequence (z n ) n N D \ {z 0 } such that z n z 0 D as n. Show that f 0. Hint: Start by defining N and N. Next, show that N N; that N is open and closed in D and deduce that N = D. Let f : D C be holomorphic and suppose that f : D R has a maximum z 0 D. Show that f is constant. 10.) One 1 of your suggestions, see Question 6 on Exercise sheet 6. (a) Define f(z) for f continuous and : [a, b] D a piecewise regular path. 1 Well, the only one! 6

7 (b) State the three properties of a path integral (no proof required). (c) State and prove the M L-inequality. (d) Let C be a simple closed contour and z 0 I a point in its interior. Prove that there is a ϱ > 0 s.t. (z z 0 ) k < 1 ϱ L. k Disclaimer: This is only selection!! In particular, it is neither complete nor will the upcoming exam be simply a subset of these questions. Furthermore: I will not provide answers 2 to these questions! 2 Find them yourself! And swap yours with some other person s! 7